Finite Group Databasebeta

← #27272572

Subgroups — #27272572

Order 13568.

562 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283064C211yesmaximal, characteristic
2#20274472C211yesmaximal, characteristic
3#20274407C211yesmaximal, characteristic
4#20272541C211yesmaximal, characteristic, complemented
5#20272540C211yesmaximal, characteristic, complemented
6#20272540C211yesmaximal, characteristic, complemented
7C53×#1272C211yesmaximal, characteristic, complemented
8C22×#11719912—21no—
9C2×#13133215C411yescharacteristic
10C2×#13133197C411yescharacteristic
11C2×#13133193C411yescharacteristic
12C2×C42×D53C2211yescharacteristic
13C2×#13133129—21no—
14C2×C8×Dic53—21no—
15#16810529C2211yescharacteristic
16#16810520—21no—
17#16810520—21no—
18#16810492C411yescharacteristic
19#16810482—21no—
20#16810482—21no—
21#16810378C2211yescharacteristic
22#16810378C2211yescharacteristic
23#16810377—21no—
24C53×#391C2211yescharacteristic
25C53×#336C2211yescharacteristic
26C53×#336C2211yescharacteristic
27C22×Dic106C2×C411yescharacteristic
28C22×C4×D53C2×C411yescharacteristic
29C22×C4×D53D411yescharacteristic
30C22×C4×D53D411yescharacteristic
31C2×#11719912—24no—
32C2×#11719912—42no—
33C2×#11719912—42no—
34C22×#88322—21no—
35C22×#88322—21no—
36#13133215—42no—
37#13133197—42no—
38#13133193—42no—
39C2×C4⋊Dic53—21no—
40C2×C4⋊Dic53C2×C411yescharacteristic
41C2×#11719919—21no—
42C2×#11719919—21no—
43C2×#11719919—21no—
44C2×#11719919—21no—
45C2×#11719919—21no—
46C2×#11719921—21no—
47C2×#11719921—21no—
48C2×C4×Dic53C2×C411yescharacteristic
49C2×C4×Dic53D411yescharacteristic
50C2×C4×Dic53D411yescharacteristic
51C42×D53—42no—
52#13133129—24no—
53C8×Dic53—42no—
54#13133124—21no—
55#13133124—21no—
56C22×Q8×C53C2×C411yescharacteristic
57C22×C8×C53—21no—
58C22×C8×C53—21no—
59C2×C4⋊C4×C53C2×C411yescharacteristic
60C2×C42×C53C2311yescharacteristic
61#95×C53—21no—
62#95×C53—21no—
63C23×D53C22⋊C411yescharacteristic
64C2×Q8×C53—21no—
65C2×Q8×C53—21no—
66C2×Q8×C53—41no—
67C2×Q8×C53—41no—
68C22×C4×C53—21no—
69C22×C4×C53—21no—
70C22×C4×C53C22×C411yescharacteristic
71C22×C4×C53C2×D411yescharacteristic
72C22×C4×C53C2×D411yescharacteristic
73C22×Dic53—21no—
74C22×Dic53—21no—
75C22×Dic53C22⋊C411yescharacteristic
76C22×Dic53C22⋊C411yescharacteristic
77C22×Dic53C22⋊C411yescharacteristic
78C2×C4×D53—22no—
79C2×C4×D53—22no—
80C2×C4×D53—21no—
81C2×C4×D53—21no—
82C2×C4×D53—21no—
83C2×C4×D53—21no—
84C2×C4×D53—22no—
85C2×C4×D53—22no—
86C2×C4×D53—41no—
87C2×C4×D53—41no—
88C2×C4×D53SD1612yes—
89C2×C4×D53Q1612yes—
90C2×Dic106—21no—
91C2×Dic106—21no—
92C2×Dic106—41no—
93C2×Dic106—41no—
94C2×C8×C53—22no—
95C2×C8×C53—22no—
96C2×C8×C53—41no—
97C2×C8×C53—41no—
98C2×C8×C53—41no—
99C2×C8×C53—41no—
100C4⋊C4×C53—41no—
101C42×C53—41no—
102#11719921—42no—
103#11719921—42no—
104C4⋊Dic53—42no—
105C4⋊Dic53—41no—
106#11719919—42no—
107#11719919—42no—
108#11719919—42no—
109#11719919—42no—
110#11719919—42no—
111C4×Dic53—22no—
112C4×Dic53—22no—
113C4×Dic53—41no—
114C2×#88322—22no—
115C2×#88322—22no—
116C2×#88322—41no—
117C2×#88322—41no—
118C2×#88322—41no—
119C2×#88322—41no—
120#11719912—48no—
121C23×C53C2×C22⋊C411yescharacteristic
122C22×D53—21no—
123C22×D53—21no—
124C22×D53—21no—
125C22×D53—21no—
126C22×D53—21no—
127C22×D53—21no—
128C22×D53#10312yes—
129Q8×C53—81no—
130Q8×C53—81no—
131C2×C4×C53—21no—
132C2×C4×C53—21no—
133C2×C4×C53—21no—
134C2×C4×C53—21no—
135C2×C4×C53—21no—
136C2×C4×C53—21no—
137C2×C4×C53—41no—
138C2×C4×C53—41no—
139C2×C4×C53—41no—
140C2×C4×C53—41no—
141C2×C4×C53—41no—
142C2×C4×C53—41no—
143C2×C4×C53#13111yescharacteristic
144C2×C4×C53#13111yescharacteristic
145C2×C4×C53C2×SD1611yescharacteristic
146C2×C4×C53C2×Q1611yescharacteristic
147C2×C4×C53#13711yescharacteristic
148C2×C4×C53#13711yescharacteristic
149C2×Dic53—21no—
150C2×Dic53—21no—
151C2×Dic53—21no—
152C2×Dic53—21no—
153C2×Dic53—41no—
154C2×Dic53—41no—
155C2×Dic53—41no—
156C2×Dic53—41no—
157C2×Dic53—41no—
158C2×Dic53—41no—
159C2×Dic53#9912yes—
160C2×Dic53#10012yes—
161C2×Dic53#10312yes—
162C2×Dic53#10412yes—
163C2×Dic53#10412yes—
164C4×D53—24no—
165C4×D53—24no—
166C4×D53—42no—
167C4×D53—42no—
168C4×D53—42no—
169C4×D53—42no—
170Dic106—81no—
171Dic106—81no—
172C8×C53—42no—
173C8×C53—42no—
174#88322—42no—
175#88322—42no—
176#7687—531nomaximal, Sylow
177C22×C53#40811yescharacteristic
178C22×C53C2×#9911yescharacteristic
179C22×C53C2×#10011yescharacteristic
180C22×C53C2×#10311yescharacteristic
181C22×C53#41911yescharacteristic
182C22×C53C2×#10411yescharacteristic
183C22×C53C2×#10411yescharacteristic
184C2×D53—22no—
185C2×D53—22no—
186C2×D53—22no—
187C2×D53—41no—
188C2×D53—41no—
189C2×D53—41no—
190C2×D53—41no—
191C4×C53—21no—
192C4×C53—21no—
193C4×C53—21no—
194C4×C53—21no—
195C4×C53—21no—
196C4×C53—21no—
197C4×C53—81no—
198C4×C53—81no—
199Dic53—22no—
200Dic53—22no—
201Dic53—81no—
202Dic53—81no—
203Dic53#32814yes—
204#2857—531no—
205C4×C22⋊C8—531no—
206#1728—531no—
207#1272—531no—
208#1272—531no—
209#1272—531no—
210#1272D5311yescharacteristic, complemented
211C2×C53#147111yescharacteristic
212C2×C53#148511yescharacteristic
213C2×C53#149311yescharacteristic
214C2×C53C2×#32811yescharacteristic
215C2×C53#150811yescharacteristic
216C2×C53#152511yescharacteristic
217C2×C53#154811yescharacteristic
218D53—42no—
219C22×M16—1061no—
220C2×C4⋊Q8—531no—
221C2×C22⋊Q8—531no—
222C2×C22⋊Q8—531no—
223C22×C42—531no—
224C2×C22⋊C8—1061no—
225C2×C4×C8—1061no—
226#391—531no—
227#391C2×D5311yescharacteristic
228#389—1061no—
229#389—1061no—
230#384—531no—
231#380—1061no—
232#380—1061no—
233#336—531no—
234#336—531no—
235#336—1061no—
236#336C2×D5311yescharacteristic
237#336C2×D5311yescharacteristic
238C53#768711yescharacteristic, complemented, Sylow
239C22×Q8—531no—
240C22×Q8C4×D5311yescharacteristic
241C23×C4—531no—
242C23×C4—531no—
243C23×C4—531no—
244C2×M16—1064no—
245C2×M16—2122no—
246C2×M16—2122no—
247C22×C8—21no—
248C22×C8—21no—
249C22×C8—1061no—
250C22×C8—1061no—
251C4⋊Q8—2122no—
252C22⋊Q8—2122no—
253C22⋊Q8—2122no—
254C2×C4⋊C4—531no—
255C2×C4⋊C4—1061no—
256C2×C4⋊C4—1061no—
257C2×C4⋊C4—1061no—
258C2×C4⋊C4—1061no—
259C2×C4⋊C4—1061no—
260C2×C4⋊C4—1061no—
261C2×C4⋊C4C4×D5311yescharacteristic
262C2×C22⋊C4—1061no—
263C2×C22⋊C4—1061no—
264C2×C42—531no—
265C2×C42—531no—
266C2×C42—531no—
267C2×C42—2122no—
268C2×C42C22×D5311yescharacteristic
269C22⋊C8—1064no—
270C4×C8—2122no—
271#95—21no—
272#95—21no—
273#95—1061no—
274#95—1061no—
275C24—531no—
276C2×Q8—21no—
277C2×Q8—21no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×Q8—1061no—
281C2×Q8—1061no—
282C2×Q8—2121no—
283C2×Q8—2121no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—531no—
287C22×C4—531no—
288C22×C4—531no—
289C22×C4—532no—
290C22×C4—532no—
291C22×C4—1062no—
292C22×C4—1062no—
293C22×C4—1062no—
294C22×C4—1062no—
295C22×C4—1061no—
296C22×C4—1061no—
297C22×C4—1061no—
298C22×C4—1061no—
299C22×C4—1061no—
300C22×C4—1061no—
301C22×C4—2121no—
302C22×C4—2121no—
303C22×C4C2×C4×D5311yescharacteristic
304C22×C4D4×D5311yescharacteristic
305C22×C4D4×D5311yescharacteristic
306M16—2128no—
307C2×C8—22no—
308C2×C8—22no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—1062no—
314C2×C8—1062no—
315C2×C8—2121no—
316C2×C8—2121no—
317C2×C8—2121no—
318C2×C8—2121no—
319C4⋊C4—41no—
320C4⋊C4—2121no—
321C4⋊C4—2122no—
322C4⋊C4—2122no—
323C4⋊C4—2122no—
324C4⋊C4—2122no—
325C4⋊C4—2122no—
326C4⋊C4—2122no—
327C22⋊C4—2122no—
328C22⋊C4—2122no—
329C42—41no—
330C42—1062no—
331C42—1062no—
332C42—2121no—
333C23—532no—
334C23—1061no—
335C23—1061no—
336C23—1061no—
337C23—1061no—
338C23—1061no—
339C23—1061no—
340C23C22⋊C4×D5311yescharacteristic
341Q8—81no—
342Q8—81no—
343Q8—4241no—
344Q8—4241no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—532no—
358C2×C4—532no—
359C2×C4—532no—
360C2×C4—532no—
361C2×C4—532no—
362C2×C4—1064no—
363C2×C4—1064no—
364C2×C4—1061no—
365C2×C4—1061no—
366C2×C4—1061no—
367C2×C4—1061no—
368C2×C4—2122no—
369C2×C4—2122no—
370C2×C4—2122no—
371C2×C4—2122no—
372C2×C4—2121no—
373C2×C4—2121no—
374C2×C4—2121no—
375C2×C4—2121no—
376C2×C4—2121no—
377C2×C4—2121no—
378C2×C4#1313322311yescharacteristic
379C2×C4#1313322511yescharacteristic
380C2×C4SD16×D5311yescharacteristic
381C2×C4Q16×D5311yescharacteristic
382C2×C4#1313324611yescharacteristic
383C2×C4#1313324911yescharacteristic
384C8—42no—
385C8—42no—
386C8—2122no—
387C8—2122no—
388C22—1062no—
389C22—1062no—
390C22—1062no—
391C22—2121no—
392C22—2121no—
393C22—2121no—
394C22—2121no—
395C22#1681058811yescharacteristic
396C22#99×D5311yescharacteristic
397C22#100×D5311yescharacteristic
398C22#103×D5311yescharacteristic
399C22#1681063011yescharacteristic
400C22#104×D5311yescharacteristic
401C22#104×D5311yescharacteristic
402C4—21no—
403C4—21no—
404C4—21no—
405C4—21no—
406C4—21no—
407C4—21no—
408C4—81no—
409C4—81no—
410C4—534no—
411C4—1062no—
412C4—1062no—
413C4—4241no—
414C4—4241no—
415C2—2122no—
416C2#2027307311yescharacteristic
417C2#2027313411yescharacteristic
418C2#2027316811yescharacteristic
419C2#328×D5311yescharacteristic
420C2#2027323311yescharacteristic
421C2#2027330511yescharacteristic
422C2#2027339411yescharacteristic
subgroupG/H
centerC23C22⋊C4×D53
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1272C2
socleC23×C53C2×C22⋊C4
prime radicalC23×D53C22⋊C4
cosocleC2×C42×C53C23
hypercenter#1272D53
hypocenterC53#7687
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1272

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. #20272541C2semidirect
5. #20272540C2semidirect
6. #20272540C2semidirect
7. C53×#1272C2semidirect
210. #1272D53semidirect
238. C53#7687semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#202830642
#202744722
#202744072
#202725412
#202725402
C53×#12722
#768753